Two Error Components Model

In 2ECM there is no time-specific error component. Thus the model is a special case of 3ECM obtained by putting o = 0. This model was first used in econometric applications by Balestra and Nerlove (1966). However, because in the Balestra-Nerlove model a lagged endogenous variable is included among regressors, which causes certain additional statistical problems, we shall discuss it separately in Section 6.6.3.

In matrix notation, 2ECM is defined by

y = X0 + u



u = 1^ + e.


The covariance matrix ft = £iiu’ is given by

ft = a* A + oJIjvt,


and its inverse by

n~l=±(lNT-yl A),


where у, = а*/{а + Та*) as before.

We can define GLS and FGLS as in Section 6.6.1. We shall discuss the estimation of y, required for FGLS later. The asymptotic equivalence of GLS and FGLS follows from Fuller and Battese (1974) because they allowed for the possibility of aj = 0 in their proof. The transformation estimator of fi can be defined as in (6.6.9) and (6.6.12) except that here the transformation matrix Q is given by

Q = I – T~lA. (6.6.22)

The asymptotic equivalence between GLS and the transformation estimator can be similarly proved, except that in the 2ECM model – JN(Дx — fi0) and qo — fi0) have the same limit distribution as SJLj fii/yfN regardless of the way N and T go to °°.

Following Maddala (1971), we can give an interesting interpretation of GLS in comparison to the transformation estimator. Let L be as defined in Section 6.6.1 and let F be an NT X N(T — 1) matrix satisfying F’L = 0 and F’F = I. Then (6.6.18) can be equivalently written as the following two sets of regres­sion equations:

7-!/2L’y = r-1/2L’X/? + ?/,


F’y = F ‘Xfi + ъ,


where Ethtfi = (a* + Ta*)IN, Etj2if2 = and Ertf2 = 0. Maddala

calls (6.6.23) the between-group regression and (6.6.24) the within-group regression. The transformation estimator can be interpreted as LS applied
to (6.6.24). (Note that since the premultiplication by F’ ejiminates the vector of ones, P0 cannot be estimated from this equation.) GLS fiG can be interpreted as GLS applied to (6.6.23) and (6.6.24) simultaneously. Because these equa­tions constitute the heteroscedastic regression model analyzed in Section 6.5.2, GLS has the following simple form:

jiG = (X’PX + cX’MX)-1(X’Py + cX’My), (6.6.25)

where c = (a2 + Tcft/o2, P = r-1LL’, and M = I – P. To define FGLS, c may be estimated as follows: Estimate a + То% by the LS estimator of the variance obtained from regression (6.6.23), estimate of by the LS estimator of the variance obtained from regression (6.6.24), and then take the ratio.

As we noted earlier, (6.6.23) and (6.6.24) constitute the heteroscedastic regression model analyzed in Section 6.5.2. Therefore the finite-sample study of Taylor (1978) applied to this model, but T aylor (1980) dealt with this model specifically.

Next, following Balestra and Nerlove (1966), we derive MLE of the model assuming the normality of u. For this purpose it is convenient to adopt the following re-parameterization used by Balestra and Nerlove: Define a2 = o + o2,p = a2/a2, and R = (1 – p)lT + рІт1’т – Then we have

£2 = <t2(I*©R),

R‘1 = d – рГЧЪ – ІР/( 1 – р+рТ)]1т1’т}, and |R| = (1 “ P)T11 + pT/{ 1 – p).

Using these results, we can write the log likelihood function as

L = -|log|Q|-^u, ft-1u (6.6.26)

NT, , NT, „ 4 Nл (, , pT

= —Ylog a —2_1°g(1 – /?) – у log ^1 +Y^p)


where we have written u for у — Xfi. From (6.6.26) it is clear that the MLE offi given p is the same as GLS and that the MLE of a2 must satisfy

Подпись: (6.6.27), ur(I® R~‘)u


Подпись: (6.6.28)

Inserting (6.6.27) into (6.6.26) yields the concentrated log likelihood function

Putting dL*/dp = 0 yields

Подпись:Подпись:image477Подпись: (6.6.30)from which we obtain

0 = –lAj_ L—- I—L—

і I

Also, using (6.6.29,) we can simplify (6.6.27) as


The MLE of fi, p, and a2 can be obtained by simultaneously solving the formula for GLS, (6.6.30), and (6.6.31).

Both Balestra and Nerlove (1966) and Maddala (1971) pointed out that the right-hand side of (6.6.30) can be negative. To ensure a positive estimate ofp, Balestra and Nerlove suggested the following alternative formula for p:

Подпись: (6.6.32)

і t

It is easy to show that the right-hand of (6.6.32) is always positive.

Maddala (1971) showed that the p given in (6.6.30) is less than 1. Berzeg (1979) showed that if we allow for a nonzero covariance between pt and e„,

the formulae for MLE are the same as those given in (6.6.30) and (6.6.31) by redefining o2 = o* + 2a^ + a and p = (a2 + lo^/a2 and in this model the MLE of p lies between 0 and 1.

One possible way to calculate FGLS consists of the following steps: (1) Obtain the transformation estimator^. (2) Define Uq = у — XfiQ; (3) Insert fig into the right-hand side of Eq. (6.6.32). (4) Use the resulting estimator of p to construct FGLS. In the third step the numerator of (6.6.32) divided by N2T2 can be interpreted as the sample variance of the LS estimator ofp( + fi0 obtained from the regression у = XJ}{ + L(p + fi01) + e.

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